LAPPEENRANTA UNIVERSITY OF TECHNOLOGY LUT School of Energy Systems
LUT Mechanical Engineering BK10A0402 Bachelor’s Thesis
PARAMETER OPTIMIZATION PROCESS OF TENDOM FROM EXPERIMENTAL DATA
JÄNTEEN PARAMETRIEN OPTIMOINTIPROSESSI KOKEELLISESTA DATASTA
Lappeenranta 7.8.2018 Erik Makkonen
Examiner and supervisor D.Sc. Marko Matikainen
ABSTRACT
Lappeenranta University of Technology LUT School of Energy Systems
LUT Mechanical Engineering Erik Makkonen
Parameter optimization process of tendon from experimental data
Bachelor’s thesis 2018
25 pages, 13 figures and 2 tables
Examiner and supervisor: D.Sc. Marko Matikainen
Keywords: parameter optimization, biomechanics, tendon, finite element, tensile test In the study material parameters are being determined for finite element (FE) model of rat Achilles tendon. A tensile test has been done with a part of real rat Achilles tendon from which time, displacement and force data is taken. Optimization code and tensile test for the tendon have been made earlier. The purpose of this thesis is to find out how the optimization code works and whether it will give usable results for FE programs.
In the code a corresponding 2D FE model is created for which a simulated tensile test is performed. The results of the simulated and experiment tensile tests are compared and model parameters are changed in the appropriate direction as many times as needed to get as close to the experiment tensile test values as possible. Finally, the code draws force-time curve that has the experiment values and the curve generated by the optimized parameters.
The study found out that the code gives close parameter values for FE model when the simulated tensile test is performed. The simulated tensile test provides a nonlinear force- time curve that is close to the experiment tensile test. The optimized parameters provided by the code can be used in the biomechanics of the FE programs, but they are probably not yet fully correct. Similar or even the same code can be used with the same experimental arrangement in other biomechanical situations. Required arrangements are time, displacement and force data and geometry of the tested piece to find the appropriate parameters for the FE programs.
TIIVISTELMÄ
Lappeenrannan teknillinen yliopisto LUT School of Energy Systems LUT Kone
Erik Makkonen
Jänteen parametrien optimointiprosessi kokeellisesta datasta
Kandidaatintyö 2018
25 sivua, 13 kuvaa ja 2 taulukkoa
Tarkastaja ja ohjaaja: TkT Marko Matikainen
Hakusanat: parametrien optimointi, biomekaniikka, jänne, elementtimenetelmä, vetokoe Työssä tutkitaan materiaaliparametrien määrittämistä rotan akillesjänteestä luodulle elementtimenetelmämallille. Oikealle rotan akillesjänteen koepalalle on tehty vetokoe, josta on otettu ylös aika, venymä ja voima. Tutkimukseen liittyvä optimointikoodi ja vetokoe jänteelle ovat tehty aikaisemmin. Työn tarkoituksena on selvittää, kuinka optimointikoodi toimii ja antaako se hyödynnettäviä tuloksia elementtimenetelmäohjelmia varten.
Koodissa luodaan oikeaa jännettä vastaava 2D elementtimenetelmämalli, jolle suoritetaan simuloitu vetokoe. Simuloidun ja oikean vetokokeen tuloksia vertaillaan ja mallin parametrejä muutetaan sopivaan suuntaan niin monta kertaa, että päästään mahdollisimman lähelle oikean vetokokeen arvoja. Lopulta koodi piirtää voima–aika-kuvaajan, jossa on alkuperäiset tilapisteet sekä koodin optimoimien parametrien avulla saatu käyrä.
Tutkimuksessa selvisi koodin antavan lähelle oikeita parametrejä elementtimenetelmämallille, kun sille suoritetaan simuloitu vetokoe. Simuloidulla vetokokeella saadaan epälineaarinen voima–aika-käyrä, joka kulkee alkuperäistä vetokoetta mukaillen. Koodin antamia optimoituja parametrejä voidaan hyödyntää elementtimenetelmäohjelmien biomekaniikan kohteissa, mutta ne eivät ole välttämättä vielä täysin oikeita. Samantapaista tai jopa samaa koodia voidaan käyttää samoilla koejärjestelyillä muissa biomekaniikan tilanteissa. Vaadittuihin koejärjestelyihin tarvitaan mitattuja tilapisteitä ja alkuperäisen kappaleen geometria, kun halutaan selvittää koodin avulla elementtimenetelmäohjelmille sopivia parametrejä.
TABLE OF CONTENTS
ABSTRACT ... 2
TIIVISTELMÄ ... 3
TABLE OF CONTENTS ... 4
LIST OF SYMBOLS AND ABBREVIATIONS ... 5
1 INTRODUCTION ... 6
2 METHODS ... 9
2.1 Material and parameters ... 11
2.2 Mesh ... 12
2.3 Optimization ... 15
3 RESULTS ... 16
4 DISCUSSION ... 18
4.1 Analyzing of results and errors ... 18
4.2 Future studies ... 21
5 SUMMARY ... 23
LIST OF REFERENCES ... 24
LIST OF SYMBOLS AND ABBREVIATIONS
Latin letters
Aα Fiber orientation vector in the reference configuration C Right Cauchy-Green strain tensor
C10 Isochoric part of Holzapfel–Gasser–Odgen model [N/mm2] D Volumetric part of Holzapfel–Gasser–Odgen model [Pa-1] Fexp Force from experiment [N]
Fmodel Force from simulated model [N]
I1 First invariant of C Jel Elastic volume ratio
k1 Fiber stiffness of Holzapfel–Gasser–Odgen model [N/mm2] k2 Fiber non-linear behavior of Holzapfel–Gasser–Odgen model N Number of families of fibers
n Number of data points U Strain energy [J]
Greek letters
κ Fiber dispersion factor
Abbreviations
FE Finite element
FEA Finite element analysis RSME Root-mean-square error
1 INTRODUCTION
Biomechanics combines mechanical physics and biological material properties. Methods of mechanics are applied to the living system to understand its structure and function, predict changes and find methods and materials for artificial intervention. (Fung 1993, p. 1;
Kreighbaum & Barthels 1996, pp. 1–3.) Biomechanical problems are for example determining the mechanical properties, environment boundaries and solving the boundary value problems analytically or numerically with help of differential equations. Materials have large deformations and their stress–strain features are usually nonlinear and history dependent. (Fung 1993, p. 11.)
Tendons attach muscles and bones to each other and forces from muscles go through tendons to move bones (Fung 1993, p. 535; Kreighbaum et al. 1996, p. 19). Mechanical properties of tendons are important to know for surgical repairs and physical treatment (Yin & Elliot 2004, p. 907). In the future it may be possible to recreate equivalent tendon from artificial materials with same mechanical properties as biological tendon.
Finite element analysis (FEA) is relevant tool in biology and it has been used to solve problems more and more (Einstein et al. 2003). Many different materials have different parameters. For working finite element (FE) model material information must be correct.
Challenge is to find these right parameters for different materials and variants. This thesis will answer to questions:
• How does the optimization process work?
• Does the optimization process give usable parameters for FEA?
This thesis contains optimization code for tendon parameters and how it works. Objective is to get similar force-time curves from experimental and simulated data. For the curves to be similar, tendon parameters have to be similar.
The code is done before with numerical computing environment (MATLAB R2017a) and it uses FE software suite (Abaqus 2018) for FEA. Goal is to show how does the optimization work with one example tendon and to get parameters for it.
In this thesis there is no comparison with different element types, but number of elements is checked that the mesh quality is high enough. One form of the strain energy potential, Holzapfel–Gasser–Ogden form is used. Only one initial guess for parameters and one tendon is under surveillance. Optimization works with two different methods and results of these two methods are shown but the main objective is to show the optimization process.
Optimization codes are great help for getting parameters for different tendons and similar hyperelastic soft tissue materials. It is a big job but it needs to be done only once. Idea is the same with different tendons and only little changes to code are possibly required.
Parameter optimization from measured data have been done before for hyperelastic materials in studies by de-Carvalho, Valente & Andrade-Campos (2011), Yin et. al. (2004) and Nguyen & Keip (2018) for example. It is known that parameters can be optimized from experimental data and this thesis shows how one process works. Parameter results can vary and exact values are hard to find. These optimization processes can help create new and better methods finding the exact values.
Reference data is from study by Eliasson et al. (2007). Tensile tests were made with part of rat Achilles tendons. The tests were about three minutes long where the tendon was pulled 20 times from 1 to 20 N (figure 1) with constant displacement rate as shown in Figure 2 with the first loading to 20 N.
Figure 1. Force-time graph from experiment.
In this thesis the first loading is under inspection and goal is to create similar force-time curve as in figure 1 the first 10 seconds where force goes to 20 N the first time. Tendon had little prestrain (table 1) and that is why displacement was not 0 when time was 0 in figure 2.
Figure 2. Displacement-time graph, first loading to 20 N.
This thesis covers the optimization code and only one tendon will be observed. The Achilles tendon will be referred as tendon 1270. Tendon’s length is 9.01 mm and radius is 0.722 mm.
In table 1 is strain data from prestrain and first upload.
Table 1. Tendon 1270 strain data.
Prestrain time [s] 1.00
Prestrain displacement [mm] 0.100
First upload time [s] 9.85
First upload displacement [mm] 1.085
Tendon was pulled from 0.100 mm displacement to 1.085 mm displacement in 9.85 seconds with a constant rate of 0.100 mm/s, as seen in figure 2. This prestrain is implemented in FEA.
2 METHODS
Optimization code creates mesh from tendon geometry and user input for number of elements. Initial guess for parameters and mesh data are written in input file for FEA solver that gives back time and reaction forces from where force-time curve is made. From experiment data another force-time curve is made and these two curves are interpolated to match the datasets exactly. Function being minimized is:
𝑓 = 1
𝑛∑𝑛𝑡=1(𝐹𝑒𝑥𝑝,𝑡− 𝐹𝑚𝑜𝑑𝑒𝑙,𝑡)2 (1)
In equation 1 Fexp is force from experiment data, Fmodel is force from simulated data and n is number of data points.
Termination tolerance on the function value is 10-4 and termination tolerance on parameter vector is also 10-4. Maximum number of iterations allowed is 1000. FE program’s subroutine creates text file after simulated tensile test where is time and reaction force data that code reads. Optimization process flowchart is presented in Figure 3.
After getting optimal parameters average error from measured reference data and simulated data can be calculated with root-mean-square error (RSME):
𝑅𝑆𝑀𝐸 = √1
𝑛∑𝑛𝑡=1(𝐹𝑒𝑥𝑝,𝑡− 𝐹𝑚𝑜𝑑𝑒𝑙,𝑡)2 (2)
Equation 2 is same as a square root of the function that is minimized in equation 1.
Figure 3. Optimization process flowchart.
From figure 3 can be seen that two sets of data are passed to minimizing algorithm, reference data and simulated data. After getting function value algorithm checks is tolerances are reached, if they are optimization is completed and if they are not algorithm gives new parameters that are written in new input file. FE program simulates tensile test with new parameters and again two sets of data is passed to the algorithm. This cycle continues as many times as needed until the tolerances are reached.
2.1 Material and parameters
Material type is anisotropic hyperelastic. Strain energy potential form is Holzapfel–Gasser–
Ogden form (Holzapfel, Gasser & Odgen 2000; Gasser, Odgen & Holzapfel 2006) with two local directions.
𝑈 = 𝐶10(𝐼̅1− 3) +1
𝐷((𝐽𝑒𝑙)2−1
2 − ln 𝐽𝑒𝑙) + 𝑘1
2𝑘2∑𝑁𝑎=1(𝑒[𝑘2〈𝐸̅𝛼〉2]− 1) (3) where
𝐸̅𝛼 ≝ 𝜅(𝐼̅1− 3) + (1 − 3𝜅)(𝐼4(𝛼𝛼)− 1) (4)
D is volumetric part, in other words volume-changing, U is the strain energy, N is the number of families of fibers (maximum of 3), 𝐼̅1 is the first invariant of 𝐂̅, the right Cauchy-Green strain tensor, the elastic volume ratio is 𝐽𝑒𝑙 and 𝐼4(𝛼𝛼) are pseudo-invariants of 𝐂̅ and 𝐀𝜶, that is the fiber orientation vector in the reference configuration. Parameters that are being optimized in equation 3 are:
• C10, isochoric part that is stress-like parameter and tries to keep tendon’s volume in its original value
• k1, stress-like parameter that relates to the stiffness of fibers,
• k2, dimensionless parameter that relates to non-linear behavior of fiber
• κ, describes the level of dispersion in the fiber directions. If κ is 0, there is no dispersion and the fibers are all aligned and if κ is 1/3 material becomes isotropic and the fibers are distributed randomly (Abaqus 2014). In Figure 4 is shown orientation of fibers. Isotropic distribution is a sphere, the transversely isotropic distributions are bone-like surfaces and aligned fibers are infinitely long line (Gasser et. al. 2006).
Figure 4. Orientation of the fibers (Gasser et. al. 2006).
Initial guess for parameters are: C10 = 5.56247·10-6 N/mm2, k1 = 6.020748·102 N/mm2, k2 = 2.542301·10-4 and κ = 0.306467.
When using elements with plane-stress formulation compressibility will be ignored (D is 0 in Abaqus) and tendon is fully isochoric (volume stays the same). This means that compressibility term in equation 3 is suppressed and the strain potential energy form is in equation 5.
𝑈 = 𝐶10(𝐼̅1− 3) + 𝑘1
2𝑘2∑𝑁𝑎=1(𝑒[𝑘2〈𝐸̅𝛼〉2]− 1) (5)
2.2 Mesh
Model is 2D part of the tendon. In Figure 5 can be seen transparent cylinder that is the part of the tendon that was in the experiment and in blue mesh the part that is simulated with FE software. Mesh will be checked that it has enough elements. In figure 5 there is only 8 elements to show where the mesh is located.
Figure 5. 3D view of tendon and mesh.
When creating mesh code takes four input arguments, tendon’s radius, tendon’s length, number of elements in radial direction and number of elements in longitudinal direction.
Code creates mesh with even spaces and generates element connectivity and node location data files for FE software. It is important that mesh quality is high enough to give correct answers. This is tested with creating model with different number of elements and results are analyzed. When adding elements does not change the answer mesh quality is good enough. Another way would be to compare results to analytical results but in this case, there are none. In Figure 6 is results with four different number of elements. They are number of elements in global x-direction times number of elements in global y-direction.
Figure 6. Force-time curves with different number of elements.
Tendon model is simple and only little differences that can be seen in Figure 6 when using 1 element versus 8, 48 or 108 elements. Only last two seconds had difference and first eight seconds were the same with every case. More details about answers will be later on in the thesis. Tendon 1270 mesh will have two elements in radial direction and 24 elements in longitudinal direction. Created mesh can be seen in Figure 7 with node indexes and boundary conditions.
Figure 7. Mesh and node indexes. Boundary conditions.
FEA make use of symmetry and only radius width is needed for the model. The left side, nodes from one to 73, cannot move in global x-direction but can in global y-direction.
Bottom edge, nodes from one to three, is fixed. When elements are generated nodes are being linked together. Generating occurs anticlockwise starting from bottom left. The first element in Figure 7 is from nodes 1, 2, 5, 4 in that order, the second element is from nodes 2, 3, 6, 5 and the third 4, 5, 8, 7. Nodes data have the location and element data have the node indexes.
Element type is four-node bilinear constant pressure hybrid element (CAX4H in Abaqus).
Element type has two degrees of freedom per node, the global x-direction and the global y- direction. Element type shows stress or displacement without twist. Data is given in global directions and the first coordinate must be greater or equal to zero. (Abaqus 2014.)
2.3 Optimization
Optimization starts with creating input file for FEA solver that has all information needed, nodes and elements, element type, boundaries, material information with initial guess for parameters that are being optimized and the test style where prestrain (table 1) have been taken into account. FEA solver’s subroutine creates text file where are time and reaction forces. Forces are per node and because in this case there are three top nodes, every set of three forces are needed to be added together. Reference data and model data are interpolated because their datasets must match exactly for comparable results. Function that is being minimized is in equation 1. It is calculated and code checks in tolerances are met. If they are not, new parameters are passed to the new input file and FEA done again as many times as needed for tolerances to be met.
There are two different methods for optimization. The first method is interior point method (Forsgren, Gill & Wright 2002) for nonlinear optimization (fmincon in MATLAB). Method has boundaries that are set. C10 is greater than or equal to 10-8 and less than or equal to 100, k1 is greater than or equal to 0 and less than or equal to 100 000, k2 has the same boundaries as k1 and κ is greater than or equal to 0 and less than or equal to 1/3. The second (fminsearch in MATLAB) is Nelder–Mead simplex direct search method (Lagarias et al. 1998). Method is unconstrained derivative-free method but code checks that there are no negative parameter values and κ is positive or zero and less than or equal to 1/3.
3 RESULTS
Result force-time curves are in Figure 8 (interior point method) and Figure 9 (Nelder-Mead simplex direct search method). Parameters, function values and RMSE are in table 2.
Figure 8. Force-time curves with experimental and optimized data with interior point method (fmincon in MATLAB).
Figure 9. Force-time curves with experimental and optimized data with Nelder-Mead simplex direct search (fminsearch in MATLAB).
Table 2. Parameter, function value and RMSE results.
Interior point method (fmincon in MATLAB)
Nelder–Mead simplex direct search method
(fminsearch in MATLAB)
C10 0.0094 4.41·10-6
k1 596.99 709.51
k2 9.3186 2.64·10-4
κ 0.3114 0.3149
Function value 0.212 0.183
RMSE 0.460 0.428
Interior point method took 27 iterations (Figure 10) and Nelder–Mead simplex direct search method took 119 iterations.
Figure 10. Function values and iterations from interior point method.
4 DISCUSSION
Optimization gives parameters with satisfying accuracy and resulting force-time curves are close to each other. Force-time curves are similar with two optimization methods but there is some difference with parameter values. In this chapter results are being analyzed and the differences are investigated. Finally, some suggestions for future studies are presented.
4.1 Analyzing of results and errors
There are some differences in parameters with the two optimization methods. From table 2 can be seen that C10 and k2 are parameters that are not close to each other but k1 and κ are at close range. Force-time curves are though very similar (figures 8 and 9). Experimental curve seems to have linear properties starting somewhere between 5 and 10 N while having nonlinear properties at the start. Simulated curves are fully nonlinear.
From Figure 10 it can be seen that with 2 iterations function that was being minimized is pretty close to the result already. Nelder–Mead simplex direct search took much longer to get optimization done, but it was a fraction closer with minimizing function in equation 1.
RMSE is with both methods excellent. In Figure 11 is tendon at the start and after 20 N load.
Figure 11. Tendon at the start and after 20 N load. Screencap from Abaqus 2018.
From figure 11 can be seen that tendon does not lose width almost at all while stretching over 1 mm, 12 % of its length. In figure 12 is shown a close up of the top where both cases are on top of each other, at the start is only shown with wireframe and 20 N load is shown with green.
Figure 12. Close up of the top with the tendon at the start (wireframe) and 20 N load tendon (green) on top of each other. Screencap from Abaqus 2018 with editing to make lines clearer.
From figure 12 close up can be seen that there is little shrinkage in width. When checking node coordinates, node 75 (top right node) moves 0.04 mm from right to left. Total area went from 6.51 mm2 to 6.88 mm2. When calculating tendon’s volumes from radius and height they are the same, 14.8 mm3, as it should be because compressibility term was ignored.
Optimization works if the initial guess for parameters are somewhere close to the answer. It is possible to get right parameters without knowing but takes lots of work and time to change initial parameters to the right direction. Now there is still question whether C10 and k2 are closer to the right answer with interior point method or with Nelder–Mead simplex search method?
This was tested with changing one parameter at the time. At first simulated tensile test was made with rounded values from interior point method: C10 = 0.01 N/mm2, k1 = 600 N/mm2,
k2 = 9 and κ = 0.311. For every parameter simulated tensile test is made with one or two lower and greater parameter values when the rest of parameters stays the same. Results are in Figure 13.
Figure 13. Force-time curves with different parameter values.
In Figure 13 orange curve is the same with every graph. In C10 and k2 cases the orange curve is under the smaller value parameter blue curve. With C10 parameter there is no difference with 0.000001 and 0.01 values. When C10 > 1 it has little effect on force-time curve and when C10 = 10 can be seen that C10 has effect on the prestrain part too. K1 is the fiber stiffness parameter and has simple effect on curves. The smaller k1 is less force is needed and vice versa. Nonlinear behavior of curve stays the same.
K2 is the nonlinear part that shows very well. Very little difference with 0.0000001 and 100 but after that it starts to matter more. K2 does not have effect on the prestrain part, only after around 5 N force nonlinear part starts showing more. Also explains the difference with the two optimization methods and from Figure 13 can be seen that with both k2 values the curves are the same. κ has a big difference with only little changes. The smaller κ is more force is
needed and κ is only parameter that has this effect on the tendon, more force is needed when other three parameter values are bigger.
From results both optimization methods give equally good values for C10 and k2 and when interior point method has lower k1 value κ is also lower that makes the curves almost the same. It may be possible that there are more good parameter values if both k1 and κ are lower because when κ is lower more force is needed but when k1 is lower less force is needed.
4.2 Future studies
For the future change the 2D model to a 3D model and test how much effect that have on the parameters and force-time curves. With 2D model fully incompressible behavior is assumed and with 3D model it is possible to take material compressibility into calculations, D in equation 3. Tendons are usually assumed to be incompressible for numerical reasons, but study by Suydam & Buchanan (2014, p. 1808) found out that human Achilles tendons are incompressible so even with 3D model and compressibility parameters probably will not change. 3D model can still have effect on the other parameters that were optimized with 2D model. Also, different element types are available that can be used.
Tendons are complex systems and parameters with this type of testing can result very different curves with other tests. It is possible that in the future there will be better strain energy potential form than Holzapfel–Gasser–Ogden form that gives more accurate parameters. There is promising looking new strain energy from proposed by Shearer (2015, p. 295) that seems to give better stress-strain curves than Holzapfel–Gasser–Odgen form in his study. For now, parameters seem to be good even though C10 and k2 had big difference with the two optimization methods, both methods give similar force-time curves and in Figure 13 can be seen that values are equally correct.
It would be possible to conduct a new experiment with different forces and test FE model with optimized parameters, see how well they match. Now on after time is 9 seconds simulated curve has distinctly steeper slope than experimental curve. What would happen if the test was two or three seconds longer?
Optimization code can be used for similar cases where tendon’s or other similar material’s length and radius is known and experiment data available. Process itself is not complicated after the code is done. The biggest challenge is to get initial guess for parameters close enough for optimization work correctly.
5 SUMMARY
FEA is a relevant tool help solving biomechanical problems and experimental data is used to compare FEA results. Challenge is to find matching parameters for FE model that has the same behavior as real material. This thesis goes through one optimization code process with one example tendon and shows how close is possible to real values when creating simulated tensile test from 1 to 20 N with constant displacement rate of 0.1 mm/s.
How does the optimization process work? We have an optimization tool that gets tendon parameters from experimental data and original tendon geometry. 2D model for FEA is created with four-node bilinear constant pressure hybrid elements and right boundary conditions, bottom nodes are fixed and left side x coordinates are bound. FE program simulates tensile test and data from experiment and simulation is compared. Minimizing and changing parameters (C10, k1, k2, κ) are done as many times as needed for function and parameter value tolerances to be met. After optimization force-time curves are made where we can see how close values from experiment and simulation are.
Does the optimization process give usable parameters for FEA? Optimization was done with two different methods and they gave good function values. There is some difference with two parameters (C10, k2), they are not close to each other but as seen in discussion chapter those differences do not matter. The other two (k1, κ) were close and the force-time curves were very similar. RSME was calculated and it resulted high accuracy with both methods.
There is still some difference with linearity and nonlinearity between simulated and experimental data. Experiment values have some linearity starting from somewhere between 5 and 10 N while simulated curves are nonlinear. The parameters are however good to use for now, probably in the future there will be some better energy strain potential form that gives even closer parameters.
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